An on-line graph coloring algorithm with sublinear performance ratio

One of the simplest heuristics for obtaining a proper coloring of a graph is the Fii-Fit algorithm: Fix an arbitrary ordering of the vertices and, using the positive integers as the color set, assign to each successive vertex the least integer possible (keeping the coloring proper). This is an example of an on-line algorithm for graph coloring. In the on-line model, a graph is presented one vertex at a time. Each new vertex is given together with all edges joining it to previous vertices. An on-line coloring algorithm assigns a color to each vertex as it is received and once assigned, the color cannot be changed. The performance function, pA(n), of an on-line algorithm A is the maximum over all graphs G on n vertices of the ratio of the number of colors used by A to color G to the chromatic numbers of G. The First-Fit algorithm has performance function n/4. We exhibit an algorithm with sublinear performance function.